Examples of differential cohomology in cohesive $\infty$ topos

Question

I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.

The formulation of differential cohomology in certain cohesive $\infty$ topos seems, to me, to be an incredibly powerful generalization of the concepts surrounding classical Chern-Weil theory. I have, for a while, been curious as to what forms differential cohomology may take in topos other than the typical one modeled on cartesian spaces (or differentiable manifolds).

Question: Can someone provide some examples of what form differential cohomology may take in topos modeled on different local data? More precisely, what does the differential cohomology diamond look like in these other topos?

To give the beginning of a possible example, suppose we take as our starting point the p-adics $\mathbb{Q}_p$ instead of the reals and form the cohesive site whose objects are $(\mathbb{Q}_p)^n$, $n\in \mathbb{N}$ and morphisms are $p$-adic smooth maps (for example locally constant when the support is compact). If I'm not mistaken, we should be able to form the differentially cohomology diamond in the resulting $\infty$ topos. Can we identify the components of this diamond with something more familiar from the world of number theory?

Answer 1

If we consider just shape modality "$\Pi$" (or "ʃ") and flat modality $\flat$ (which are sufficient for the differential cohomology hexagon, then the traditional arithmetic fracture squares are the left half of the differential hexagon for a kind of pointed $E_\infty$-derived cohesive geometry modeled on Greenlees-May duality. For more on this see my talk Differential cohesion and Idelic structure.

If we consider in addition also the sharp modality $\sharp$ (which is needed to construct moduli stacks of differential cocycles) then an "exotic" model for cohesive homotopy theory is global equivariant homotopy theory. This was found by Charkes Rezk, see his writeup "Global Homotopy Theory and Cohesion".

For these "exotic" models shape is not the localization at an interval object, and hence these exotic differential cohomology theories need not satisy the fundamental theorem of calculus/Stokes theorem (nLab:Stokes theorem -- Abstract formulation in cohesion)

By the way, even if the "standard model" with interval object the real line feels very standard, there are things still to be learned here. Readers with an interest in formal logic might enjoy Mike Shulman's recent "Brouwer's fixed-point theorem in real-cohesive homotopy type theory".

A mild but important variant of the standard model over smooth manifolds which still does satisfy the integration theorem is the model over complex analytic manifolds (see at complex analytic infinity-groupoid). In here ordinary differential cohomology comes out as the theory of holomorphic $p$-gerbes with connection (which of course was what Deligne considered first, back in 1971.) This is also the context of Hopkins-Quick 12.

Then, both the model over smooth and over complex-analytic manifolds may be refined to supergeometry. When doing so, then the adjoint triple of cohesion is accompanied by a progression of two more adjoint triples, each "resolving" the former one. The second one gives what I had called "differential cohesion", which serves to axiomatized manifolds and PDE theory. The third reflects the two possible ways of projecting out bosonic formal geometry from supergeometry. This gives a second cohesive structure on supergeometric homotopy theory.

Details for this super-cohesion are in v2 of my book, some key points are in nLab:super Cartan geometry; exposition includes Modern physics formalized in Modal homotopy theory. Maybe see also the chapter Higher prequantum geometry which I am preparing for "New Spaces in Mathematics and Physics".